by Lutz Schröder and Dirk Pattinson
Abstract:
Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities.
Reference:
Lutz Schröder and Dirk Pattinson: Strong completeness of coalgebraic modal logics, In Susanne Albers, Jean-Yves Marion, eds.: 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), Leibniz International Proceedings in Informatics, pp. 673–684, Schloss Dagstuhl - Leibniz-Center of Informatics; Dagstuhl, Germany, 2009.
Bibtex Entry:
@InProceedings{SchroderPattinson09,
author = {Lutz Schr{\"o}der and Dirk Pattinson},
title = {Strong completeness of coalgebraic modal logics},
year = {2009},
editor = {Susanne Albers and Jean-Yves Marion},
booktitle = {26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009)},
publisher = {Schloss Dagstuhl - Leibniz-Center of Informatics; Dagstuhl, Germany},
series = {Leibniz International Proceedings in Informatics},
pages = {673--684},
url = {http://drops.dagstuhl.de/opus/volltexte/2009/1855},
abstract = {Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities.},
}